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Reality-constrained Minimal Yukawa Structure in SO(10) GUT

Shaikh Saad, Vasja Susič

Abstract

We investigate the minimal Yukawa sector of grand unified theories based on $\mathrm{SO}(10)$ symmetry, consisting of a Higgs structure with representations $\mathbf{10}_{\mathbb{R}}\oplus \mathbf{120}_{\mathbb{R}}\oplus\mathbf{126}$. In this framework, where $\mathbf{10}_\mathbb{R}$ and $\mathbf{120}_{\mathbb{R}}$ are real scalars, we derive the associated $\mathrm{SO}(10)$ reality conditions for their weak-doublet constituents -- both by explicit computation and an analytic reframing into a Pati-Salam-like description -- to revisit previously reported fermion mass relations. Our analysis revises these earlier results, in particular by introducing a relative sign difference between the reality constraints on the two weak doublets in $\mathbf{120}_{\mathbb{R}}$, yielding a new parameter (a magnitude) in the fermion mass relations. Our formalism is fully general and provides a systematic framework for deriving Clebsch-Gordan coefficients and implementing reality constraints for arbitrary parent-daughter representation pairs of $\mathrm{SO}(10)$ and its Pati-Salam subgroup. Incorporating these corrections, we perform an extensive numerical scan of the parameter space and find that the model successfully reproduces SM fermion masses and mixings, including recent precision measurements of solar oscillation parameters by JUNO. It accommodates both octants of $θ_{23}$ while mildly disfavoring $δ_\mathrm{PMNS}\sim (140^\circ,220^\circ)$. The model predicts a strongly hierarchical right-handed neutrino spectrum $(10^{5},10^{12},10^{15})$ GeV and a neutrinoless double beta decay parameter $m_{ββ}\sim 3$-$4$ meV, just below future experimental sensitivity. Proton decay is dominated by $p\toπ^+\overlineν$ and $p\toπ^0 e^+$, making these channels testable in upcoming experiments.

Reality-constrained Minimal Yukawa Structure in SO(10) GUT

Abstract

We investigate the minimal Yukawa sector of grand unified theories based on symmetry, consisting of a Higgs structure with representations . In this framework, where and are real scalars, we derive the associated reality conditions for their weak-doublet constituents -- both by explicit computation and an analytic reframing into a Pati-Salam-like description -- to revisit previously reported fermion mass relations. Our analysis revises these earlier results, in particular by introducing a relative sign difference between the reality constraints on the two weak doublets in , yielding a new parameter (a magnitude) in the fermion mass relations. Our formalism is fully general and provides a systematic framework for deriving Clebsch-Gordan coefficients and implementing reality constraints for arbitrary parent-daughter representation pairs of and its Pati-Salam subgroup. Incorporating these corrections, we perform an extensive numerical scan of the parameter space and find that the model successfully reproduces SM fermion masses and mixings, including recent precision measurements of solar oscillation parameters by JUNO. It accommodates both octants of while mildly disfavoring . The model predicts a strongly hierarchical right-handed neutrino spectrum GeV and a neutrinoless double beta decay parameter - meV, just below future experimental sensitivity. Proton decay is dominated by and , making these channels testable in upcoming experiments.

Paper Structure

This paper contains 18 sections, 55 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Deriving the reality conditions
  3. Group theory: preliminaries and conventions
  4. The reality-condition ambiguity in Pati-Salam language
  5. Reality constraint through an explicit $\mathrm{SO}(10)$ computation
  6. Expressing the reality constraint in Pati-Salam language
  7. Minimal Yukawa Sector of $\mathrm{SO}(10)$
  8. Parametrization of the Yukawa expressions
  9. Setup
  10. Parameter counting
  11. Fit to the SM Fermion Sector
  12. Numerical Procedure
  13. Fit Results
  14. Conclusions
  15. An overview of spinors in $\mathrm{SO}(2n)$
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Left: minimum $\chi^{2}$ for different target values of $\delta_\mathrm{PMNS}$ (solid line), and the dominant contributions to it from observables $y_{b}$, $y_{b}$ and $\sin^{2}\theta_{23}$ (dashed lines). The points on the solid curve denote the actual data, with circular/square markers denoting the $\theta_{23}$ octant in which the fit settled. For comparison we show also the inferred $\chi^{2}_{\delta}$ from the NuFIT 6.0 NUFIT global fit of experimental results, which is not included in the total $\chi^{2}$ of fits. Right: the relative volume associated to the hypersurfaces on which the best-fit points lie; computed in two different ways, see main text.
  • Figure 2: An interesting correlation $M_i (\delta_\mathrm{PMNS})$ between the heavy right-handed neutrino masses (normalized to $M_{i}(0)$) and the PMNS phase.
  • Figure 3: Left panel: MCMC results illustrating the correlation between the neutrino mixing angle $\theta_{12}$ and the mass-squared difference $\Delta m^2_{21}$. As can be seen from this plot, our results are fully consistent with the recent JUNO measurement JUNO:2025gmd. Right panel: the HPD contours for the correlation of the leptonic Dirac CP-violating phase $\delta_\mathrm{PMNS}$ and the atmospheric mixing $\sin^2\theta_{23}$, obtained from the MCMC analyses. It is to be pointed out that even though the MCMC did not explore the entire range, the model can accommodate any value $\delta_\mathrm{PMNS} \in [0,2\pi)$, see text for details.
  • Figure 5: Correlations in MCMC results in the branching ratio of $p\to \pi^{0} e^{+}$ with two other processes. Left panel: the correlation with the largest branching ratio $p\to \pi^{+} \overline{\nu}$. Right panel: the correlation with $p\to\pi^{0}\mu^{+}$, providing information about the flavor structure in the lepton sector.